Second-order energy-stable scheme and superconvergence for the finite difference method on non-uniform grids for the viscous Cahn–Hilliard equation

In this work, we construct a fully discrete scheme with finite difference method based on the staggered grids for the viscous Cahn–Hilliard equation. The constructed scheme can satisfy the unconditional dissipation law with original energy. We carry out a rigorous error analysis with superconvergence by introducing an auxiliary function depending on the chemical potential. We obtain second order accuracy in both space and time with \(l^{\infty }(0,T; H^1(\Omega ))\) $l^{\infty }(0,T;H^1(\mathrm{\Omega }))$ norm for the phase function and \(l^{2}(0,T; l^2(\Omega ))\) $l^2(0,T;l^2(\mathrm{\Omega }))$ for the chemical potential on non-uniform grids. Numerical experiments are presented to verify the theoretical results.